set.seed(123)
Homoscedastic Error Variances
For the visualization of homocedastic and hetercedastic variance, we
simulate:
\[
y \sim N \left ( -1 + 2x, 1 \right )
\]
The funnel-shaped heterocedasticy is simulated using:
\[
y \sim N \left ( -1 + 2x, \left ( 0.1 + 0.3 \left ( x + 3 \right )
\right )^{2} \right )
\]
par(mfrow = c(2, 2))
# Homecedastic variance
X <- seq(-2, 2, length.out = 300)
homecedastic_random <- function (x){
rnorm(1, mean = -1 + 2 * x, sd = 1)[1]
}
homo_y <- sapply(X, homecedastic_random)
homo_e <- (-1 + 2 * X) - homo_y
plot(X, homo_y, xlab='', ylab='', main='homocedastic variance')
abline(-1, 2, col = "red", lwd=2)
plot(X, homo_e, xlab='', ylab='', main='homocedastic variance errors')
abline(0, 0, col = "red", lwd=2)
heterocedastic_random <- function (x){
rnorm(1, mean = -1 + 2 * x, sd = 0.1 + 0.3 * (x + 3))[1]
}
hetero_y <- sapply(X, heterocedastic_random)
hetero_e <- (-1 + 2 * X) - hetero_y
plot(X, hetero_y, xlab='', ylab='', main='funnel-shaped heterocedastic variance')
abline(-1, 2, col = "red", lwd=2)
plot(X, hetero_e, xlab='', ylab='', main='funnel-shaped heterocedastic variance errors')
abline(0, 0, col = "red", lwd=2)

Example 3.1 Munich Rent Index—Heteroscedastic Variances
par(mfrow = c(1, 2))
munich_rent_url <- "https://www.uni-goettingen.de/de/document/download/64c29c1b1fccb142cfa8f29a942a9e05.raw/rent99.raw"
munich_rent_index <- read.table(
url(munich_rent_url),
header = 1,
colClasses = c(
"numeric", "numeric", "numeric",
"numeric", "factor", "factor",
"factor", "factor", "factor"
)
)
simple_reg <- lm(rent ~ area, data = munich_rent_index)
plot(munich_rent_index$area, munich_rent_index$rent, xlab = 'area in sqm', ylab = 'net rent in Euro')
abline(simple_reg, col = "red", lwd = 2)
predicted <- predict(simple_reg)
plot(munich_rent_index$area, predicted - munich_rent_index$rent, xlab = 'area in sqm', ylab = 'residuals')
abline(0, 0, col = "red", lwd = 2)

Uncorrelated Errors
Simulated autocorrelated errors with positive correlation are
generated using:
\[
y_{i} = -1 + 2 x_{i} + \epsilon_{i}
\] \[
\text{Where } \epsilon_{i} = 0.9 \epsilon_{i - 1} + \mu_{i} \wedge
\mu_{i} \sim N(0, 0.5^{2})
\]
par(mfrow = c(1, 2))
n <- 100
X <- seq(-3, 3, length.out = n)
mu <- rnorm(n, mean = 0, sd = 0.5)
epsilon <- mu[1]
for(i in seq(2, n)){
epsilon <- c(epsilon, 0.9 * epsilon[i - 1] + mu[i])
}
y_pred <- -1 + 2 * X + epsilon
plot(X, y_pred, xlab = 'x', ylab = '', main = 'observations and regression line')
abline(-1, 2, col = "red", lwd = 2)
plot(epsilon, xlab = 'i', ylab = '', main = 'time series of the errors')
abline(0, 0, col = "red", lwd = 2)

Simulated autocorrelated errors with negative correlation are
generated using:
\[
y_{i} = -1 + 2 x_{i} + \epsilon_{i}
\] \[
\text{Where } \epsilon_{i} = - 0.9 \epsilon_{i - 1} + \mu_{i} \wedge
\mu_{i} \sim N(0, 0.5^{2})
\]
par(mfrow = c(1, 2))
n <- 100
X <- seq(-3, 3, length.out = n)
mu <- rnorm(n, mean = 0, sd = 0.5)
epsilon <- mu[1]
for(i in seq(2, n)){
epsilon <- c(epsilon, -0.9 * epsilon[i - 1] + mu[i])
}
y_pred <- -1 + 2 * X + epsilon
plot(X, y_pred, xlab = 'x', ylab = '', main = 'observations and regression line')
abline(-1, 2, col = "red", lwd = 2)
plot(epsilon, xlab = 'i', ylab = '', main = 'time series of the errors')
abline(0, 0, col = "red", lwd = 2)

Mispecified model is simulated using:
\[
y_{i} = \sin(x_{i}) + x_{i} + \epsilon_{i}
\] \[
\text{Where } \epsilon_{i} \sim N(0, 0.3^{2})
\]
par(mfrow = c(2, 2))
n <- 100
X <- seq(-3, 3, length.out = n)
epsilon <- rnorm(n, mean = 0, sd = 0.3)
y <- sin(X) + X + epsilon
plot(X, y, xlab = '', ylab = '', main = 'observations and true function')
lines(X, sin(X) + X, col = "red", lwd = 2)
linear_model <- lm(y ~ X)
plot(X, y, xlab = '', ylab = '', main = 'observations and regression line')
abline(linear_model, col = "red", lwd = 2)
y_pred <- predict(linear_model, data = X)
res <- y_pred - y
plot(X, res, xlab = '', ylab = '', main = 'residuals')
abline(0, 0, col = "red", lwd = 2)

Additivity of Errors
We simulate data using:
\[
y_{i} = \exp(1 + x_{i1} - x_{i2} + \epsilon_{i})
\]
With: \[
\epsilon_{i} \sim N \left ( 0, {0.4}^{2} \right )
\]
We plot this model:
par(mfrow = c(2, 2))
n <- 50
x1 <- seq(0, 3, length.out = n)
x2 <- seq(0, 3, length.out = n)
x_grid <- expand.grid(x1 = x1, x2 = x2)
epsilon <- rnorm(n^2, mean = 0, sd = 0.4)
y <- exp(1 + x_grid$x1 - x_grid$x2 + epsilon)
plot(
x_grid$x1, y,
main = 'scatter plot: y versus x1',
xlab = 'x1',
ylab = 'y'
)
plot(
x_grid$x2, y,
main = 'scatter plot: y versus x2',
xlab = 'x2',
ylab = 'y'
)
plot(
x_grid$x1, log(y),
main = 'scatter plot: log(y) versus x1',
xlab = 'x1',
ylab = 'log(y)'
)
plot(
x_grid$x2, log(y),
main = 'scatter plot: log(y) versus x2',
xlab = 'x2',
ylab = 'log(y)'
)

Example 3.2 Supermarket Scanner Data
Data not available online.
Example 3.3 Munich Rent Index — Variable Transformation
Importing data using script:
source("import_data/munich_rent_index.R")
We do the regression model:
\[
\text{rentsqm}_{i} = \beta_{0} + \beta_{1} \cdot f(\text{area}_{i}) +
\epsilon_{i}
\]
With both \(f(\cdot)\):
\[
f(\text{area}_{i}) = \text{area}_{i}
\]
For linear model and:
\[
f(\text{area}_{i}) = \frac{1}{\text{area}_{i}}
\]
For future use we define those models:
Model 1 (\(M1\)):
\[
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \text{area}
\]
Model 2 (\(M2\)):
\[
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \frac{1}{\text{area}}
\]
For nonlinear relationship between \(\text{rentsqm}\) and \(\text{area}\). The plotted graph below are
those two models with the left panels the linear regression without the
transformation and the right panels with the inverse transformation.
par(mfrow = c(3, 2))
m1 <- lm(rentsqm ~ area, data = munich_rent_index)
summary(m1)
Call:
lm(formula = rentsqm ~ area, data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-6.9622 -1.5737 -0.1102 1.5861 9.9674
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.46883 0.12426 76.20 <2e-16 ***
area -0.03499 0.00174 -20.11 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.291 on 3080 degrees of freedom
Multiple R-squared: 0.1161, Adjusted R-squared: 0.1158
F-statistic: 404.5 on 1 and 3080 DF, p-value: < 2.2e-16
m2 <- lm(rentsqm ~ I(1 / area), data = munich_rent_index)
summary(m2)
Call:
lm(formula = rentsqm ~ I(1/area), data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-7.3963 -1.5733 -0.0921 1.5824 10.1287
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.7321 0.1084 43.65 <2e-16 ***
I(1/area) 140.1783 5.9287 23.64 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.241 on 3080 degrees of freedom
Multiple R-squared: 0.1536, Adjusted R-squared: 0.1533
F-statistic: 559 on 1 and 3080 DF, p-value: < 2.2e-16
plot_rentsqm_area <- function(title) {
plot(
munich_rent_index$area,
munich_rent_index$rentsqm,
main = title,
xlab = 'area in sqm',
ylab = 'rent per sqm'
)
}
seq_area <- seq(min(munich_rent_index$area), max(munich_rent_index$area), length.out = 100)
plot_rentsqm_area('rent per sqm vs. area')
m1_beta <- m1$coefficients
lines(seq_area, m1_beta[1] + m1_beta[2] * seq_area, col = 'red', lwd = 2)
plot_rentsqm_area('f(area) = 1/area')
m2_beta <- m2$coefficients
lines(seq_area, m2_beta[1] + m2_beta[2] * 1 / seq_area, col = 'red', lwd = 2)
plot_residuals <- function(model) {
plot(
munich_rent_index$area,
model$residuals,
main = 'residuals',
xlab = 'area in sqm',
ylab = 'residuals'
)
abline(0, 0, col = 'red', lwd = 2)
}
plot_residuals(m1)
plot_residuals(m2)
plot_av_residuals <- function(model) {
av_residuals <- tapply(
model$residuals, munich_rent_index$area,
mean
)
plot(
unique(munich_rent_index$area),
av_residuals,
main = 'average residuals',
xlab = 'area in sqm',
ylab = 'residuals'
)
abline(0, 0, col = 'red', lwd = 2)
}
plot_av_residuals(m1)
plot_av_residuals(m2)

Example 3.4 Munich Rent Index — Polynomial Regression
For polynomial regressions we adjust two different model. Of
second-order (Model 3 (\(M3\))):
\[
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \text{area} +
\beta_{2} \text{area}^{2}
\]
And third-order (Model 4 (\(M4\))):
\[
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \text{area} +
\beta_{2} \text{area}^{2} + \beta_{3} \text{area}^{3}
\]
par(mfrow = c(2, 2))
m3 <- lm(rentsqm ~ area + I(area^2), data = munich_rent_index)
summary(m3)
Call:
lm(formula = rentsqm ~ area + I(area^2), data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-7.2150 -1.5816 -0.0915 1.5869 9.9392
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.183e+01 2.684e-01 44.081 <2e-16 ***
area -1.057e-01 7.351e-03 -14.380 <2e-16 ***
I(area^2) 4.707e-04 4.758e-05 9.892 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.255 on 3079 degrees of freedom
Multiple R-squared: 0.1433, Adjusted R-squared: 0.1428
F-statistic: 257.6 on 2 and 3079 DF, p-value: < 2.2e-16
m4 <- lm(rentsqm ~ area + I(area^2) + I(area^3), data = munich_rent_index)
summary(m4)
Call:
lm(formula = rentsqm ~ area + I(area^2) + I(area^3), data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-7.4269 -1.5854 -0.1101 1.5948 10.0670
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.428e+01 5.730e-01 24.915 < 2e-16 ***
area -2.175e-01 2.432e-02 -8.946 < 2e-16 ***
I(area^2) 1.981e-03 3.167e-04 6.254 4.54e-10 ***
I(area^3) -6.105e-06 1.266e-06 -4.823 1.49e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.247 on 3078 degrees of freedom
Multiple R-squared: 0.1497, Adjusted R-squared: 0.1489
F-statistic: 180.7 on 3 and 3078 DF, p-value: < 2.2e-16
plot_rentsqm_area('rent per sqm vs. area')
m3_beta <- m3$coefficients
lines(seq_area, m3_beta[1] + m3_beta[2] * seq_area + m3_beta[3] * seq_area ^ 2, col = 'red', lwd = 2)
plot_rentsqm_area('f(area) = 1/area')
m4_beta <- m4$coefficients
lines(seq_area, m4_beta[1] + m4_beta[2] * seq_area + m4_beta[3] * seq_area ^ 2 + m4_beta[4] * seq_area ^ 3, col = 'red', lwd = 2)
plot_residuals(m3)
plot_residuals(m4)

Example 3.5 Munich Rent Index — Additive Models
We define the following aditive models. Additive model 1:
\[
\mathcal{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot
\frac{1}{\text{area}_{i}} + \beta_{2} \cdot \text{yearc}_{i}
\]
And a second one with \(\text{yearc}\) as a polinomial of degree
3:
\[
\mathcal{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot
\frac{1}{\text{area}_{i}} + \beta_{2} \cdot \text{yearc}_{i} + \beta_{3}
\cdot {\text{yearc}_{i}}^{2} + \beta_{4} \cdot {\text{yearc}_{i}}^{3}
\]
additive_m1 <- lm(rentsqm ~ I(1/area) + yearc, data = munich_rent_index)
summary(additive_m1)
Call:
lm(formula = rentsqm ~ I(1/area) + yearc, data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-7.2682 -1.4894 -0.1401 1.3935 9.0582
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -65.406008 3.351088 -19.52 <2e-16 ***
I(1/area) 119.360735 5.636182 21.18 <2e-16 ***
yearc 0.036033 0.001721 20.94 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.097 on 3079 degrees of freedom
Multiple R-squared: 0.2591, Adjusted R-squared: 0.2586
F-statistic: 538.5 on 2 and 3079 DF, p-value: < 2.2e-16
additive_m2 <- lm(rentsqm ~ I(1/area) + yearc + I(yearc ^ 2)+ I(yearc ^ 3), data = munich_rent_index)
summary(additive_m2)
Call:
lm(formula = rentsqm ~ I(1/area) + yearc + I(yearc^2) + I(yearc^3),
data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-6.9920 -1.3705 -0.1354 1.3711 8.2834
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.942e+04 2.993e+04 0.983 0.326
I(1/area) 1.296e+02 5.536e+00 23.406 <2e-16 ***
yearc -4.330e+01 4.592e+01 -0.943 0.346
I(yearc^2) 2.119e-02 2.348e-02 0.902 0.367
I(yearc^3) -3.445e-06 4.002e-06 -0.861 0.389
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.039 on 3077 degrees of freedom
Multiple R-squared: 0.3, Adjusted R-squared: 0.2991
F-statistic: 329.7 on 4 and 3077 DF, p-value: < 2.2e-16
A third model is specified using the truncated year of construction
(\(\text{yearc} - 1900\)):
munich_rent_index$yearco <- munich_rent_index$yearc - 1900
additive_m3 <- lm(rentsqm ~ I(1/area) + yearco + I(yearco ^ 2)+ I(yearco ^ 3), data = munich_rent_index)
summary(additive_m3)
Call:
lm(formula = rentsqm ~ I(1/area) + yearco + I(yearco^2) + I(yearco^3),
data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-6.9920 -1.3705 -0.1354 1.3711 8.2834
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.417e+00 4.779e-01 11.335 < 2e-16 ***
I(1/area) 1.296e+02 5.536e+00 23.406 < 2e-16 ***
yearco -9.434e-02 3.384e-02 -2.788 0.00534 **
I(yearco^2) 1.553e-03 6.728e-04 2.308 0.02105 *
I(yearco^3) -3.445e-06 4.002e-06 -0.861 0.38947
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.039 on 3077 degrees of freedom
Multiple R-squared: 0.3, Adjusted R-squared: 0.2991
F-statistic: 329.7 on 4 and 3077 DF, p-value: < 2.2e-16
As in the book, we show the effect of year of construction in the
polynomial of degree 3:
par(mfrow = c(1, 2))
beta_add_m2 <- additive_m2$coefficients
yearc_seq <- seq(min(munich_rent_index$yearc), max(munich_rent_index$yearc), length.out = 100)
plot_effect <- function(beta, year_seq, title){
plot(
year_seq,
beta[3] * year_seq + beta[4] * year_seq^2 + beta[5] * year_seq^3,
main = title,
xlab = 'year of construction',
ylab = 'effect of year of construction',
type = 'l',
col = 'darkblue',
lwd = 2
)
}
plot_effect(beta_add_m2, yearc_seq, 'effect of year of construction')
beta_add_m3 <- additive_m3$coefficients
truncate_yearc_seq <- seq(min(munich_rent_index$yearco), max(munich_rent_index$yearco), length.out = 100)
plot_effect(beta_add_m3, truncate_yearc_seq, 'effect of year of construction, coded from 18 to 96')

A new model is defined using the orthogonal polynomial of year of
construction:
munich_rent_index$areainv <- (1 / munich_rent_index$area) - mean(1 / munich_rent_index$area)
poly_year <- poly(munich_rent_index$yearco - mean(munich_rent_index$yearco), 3)
munich_rent_index$yearcc <- poly_year[, 1]
munich_rent_index$yearcc2 <- poly_year[, 2]
munich_rent_index$yearcc3 <- poly_year[, 3]
additive_m4 <- lm(rentsqm ~ areainv + yearcc + yearcc2 + yearcc3, data = munich_rent_index)
summary(additive_m4)
Call:
lm(formula = rentsqm ~ areainv + yearcc + yearcc2 + yearcc3,
data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-6.9920 -1.3705 -0.1354 1.3711 8.2834
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.11126 0.03674 193.575 <2e-16 ***
areainv 129.57166 5.53582 23.406 <2e-16 ***
yearcc 43.93838 2.07260 21.200 <2e-16 ***
yearcc2 27.53892 2.05958 13.371 <2e-16 ***
yearcc3 -1.75582 2.03998 -0.861 0.389
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.039 on 3077 degrees of freedom
Multiple R-squared: 0.3, Adjusted R-squared: 0.2991
F-statistic: 329.7 on 4 and 3077 DF, p-value: < 2.2e-16
Now we calculate the partial residuals for \(\text{area}\) and \(\text{yearc}\) define by:
\[
\hat{\epsilon}_{\text{area}, i} = \text{rentsqm}_{i} - \hat{\beta_{0}} -
\hat{f}_{2}(\text{yearc}_{i})
\]
With the effect \(f(\text{yearc}_{i})\):
\[
\hat{f}_{2}(\text{yearc}) = \beta_{2} \cdot \text{yearc}_{i} + \beta_{3}
\cdot {\text{yearc}_{i}}^{2} + \beta_{4} \cdot {\text{yearc}_{i}}^{3}
\]
And:
\[
\hat{\epsilon}_{\text{yearc}, i} = \text{rentsqm}_{i} - \hat{\beta_{0}}
- \hat{f}_{1}(\text{area}_{i})
\]
With the effect \(f(\text{yearc}_{i})\):
\[
\hat{f}_{1}(\text{yearc}) = \beta_{1} \cdot \frac{1}{\text{area}_{i}}
\]
par(mfrow = c(1, 2))
beta_add_m4 <- additive_m4$coefficients
area_seq <- seq(min(munich_rent_index$area), max(munich_rent_index$area), length.out = 100)
area_effect <- function(area){
inv_area <- 1 / area
cent_area <- inv_area - mean(inv_area)
beta_add_m4[2] * cent_area
}
yearc_effect <- function(yearc){
yearc_center <- yearc - mean(yearc)
poly_yearc <- poly(yearc_center, 3)
beta_add_m4[3] * poly_yearc[, 1] + beta_add_m4[4] * poly_yearc[, 2] + beta_add_m4[5] * poly_yearc[, 3]
}
residual_area <- munich_rent_index$rentsqm - beta_add_m4[1] - yearc_effect(munich_rent_index$yearc)
plot(
munich_rent_index$area,
residual_area,
main = 'effect of area',
xlab = 'area in sqm',
ylab = 'effect of area / partial residuals'
)
lines(area_seq, area_effect(area_seq), col = 'red', lwd = 2)
residual_year <- munich_rent_index$rentsqm - beta_add_m4[1] - area_effect(munich_rent_index$area)
plot(
munich_rent_index$yearc,
residual_year,
main = 'effect of year of construction',
xlab = 'year of construction',
ylab = 'effect of year of construction / partial residuals'
)
lines(yearc_seq, yearc_effect(yearc_seq), col = 'red', lwd = 2)

Example 3.6 Munich Rent Index — Effect Coding
We adjust the regression model with the coding values of
location:
\[
\mathbb{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot
\text{glocation}_{1} + \beta_{2} \cdot \text{tlocation}_{1}
\]
munich_rent_index$glocation <- 0
munich_rent_index$glocation[munich_rent_index$location == 2] <- 1
munich_rent_index$glocation[munich_rent_index$location == 1] <- -1
munich_rent_index$tlocation <- 0
munich_rent_index$tlocation[munich_rent_index$location == 3] <- 1
munich_rent_index$tlocation[munich_rent_index$location == 1] <- -1
cod_model <- lm(
rentsqm ~ glocation + tlocation,
data = munich_rent_index
)
summary(cod_model)
Call:
lm(formula = rentsqm ~ glocation + tlocation, data = munich_rent_index)
Residuals:
Min 1Q Median 3Q Max
-6.8564 -1.8376 -0.1074 1.7157 10.4494
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.46704 0.09638 77.477 < 2e-16 ***
glocation -0.19479 0.10445 -1.865 0.062285 .
tlocation 0.70529 0.18558 3.800 0.000147 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.426 on 3079 degrees of freedom
Multiple R-squared: 0.008867, Adjusted R-squared: 0.008223
F-statistic: 13.77 on 2 and 3079 DF, p-value: 1.109e-06
Example 3.7 Munich Rent Index — Interaction with Quality of
Kitchen
We import the new updated model of 2001, available at official page of the
dataset as all the datasets:
munich_rent_url_01 <- "https://www.uni-goettingen.de/de/document/download/e00d039e9c1f28ab404b27b0b14931f1.raw/rent98_00.raw"
munich_rent_index_01 <- read.table(
url(munich_rent_url_01),
header = 1,
colClasses = c(
"numeric", "numeric", "numeric",
"numeric", "integer", "integer",
"integer", "integer", "integer",
"integer", "numeric", "numeric",
"numeric"
)
)
summary(munich_rent_index_01)
rent_euro rentsqm_euro area yearc
Min. : 12.83 Min. : 0.1841 Min. : 20.00 Min. :1918
1st Qu.: 320.86 1st Qu.: 5.2601 1st Qu.: 52.00 1st Qu.:1939
Median : 424.12 Median : 6.8614 Median : 66.00 Median :1960
Mean : 456.94 Mean : 7.0274 Mean : 67.92 Mean :1956
3rd Qu.: 552.40 3rd Qu.: 8.7292 3rd Qu.: 82.00 3rd Qu.:1972
Max. :1837.89 Max. :17.6688 Max. :243.00 Max. :1999
glocation tlocation nkitchen pkitchen
Min. :0.0000 Min. :0.00000 Min. :0.00000 Min. :0.00000
1st Qu.:0.0000 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000
Median :0.0000 Median :0.00000 Median :0.00000 Median :0.00000
Mean :0.3907 Mean :0.02516 Mean :0.09888 Mean :0.04004
3rd Qu.:1.0000 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000
Max. :1.0000 Max. :1.00000 Max. :1.00000 Max. :1.00000
eboden year01 yearc2 yearc3
Min. :0.0000 Min. :0.0000 Min. :3678724 Min. :7.060e+09
1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:3759721 1st Qu.:7.290e+09
Median :0.0000 Median :0.0000 Median :3841600 Median :7.530e+09
Mean :0.2986 Mean :0.3257 Mean :3828362 Mean :7.493e+09
3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:3888784 3rd Qu.:7.670e+09
Max. :1.0000 Max. :1.0000 Max. :3996001 Max. :7.990e+09
invarea
Min. :0.004115
1st Qu.:0.012195
Median :0.015152
Mean :0.016904
3rd Qu.:0.019231
Max. :0.050000
And we adjust the model:
\[
\begin{matrix}
\widehat{\text{rentsqm}}_{i} = & \beta_{0} + \beta_{1}
\text{areainvc}_{i} + \beta_{2} \text{yearco}_{i} + \beta_{3}
\text{yearco2}_{i} + \beta_{4} \text{yearco3}_{i} \\
& + \beta_{5} \text{year01}_{i} + \beta_{6} \text{nkitchen}_{i} +
\beta_{7} \text{pkitchen}_{i} \\
& + \beta_{8} \text{nkitchen}_{i} \cdot \text{year01}_{i} +
\beta_{9} \text{pkitchen}_{i} \cdot \text{year01}_{i}
\end{matrix}
\]
munich_rent_index_01$areainvc <- munich_rent_index_01$invarea - mean(munich_rent_index_01$invarea)
munich_rent_index_01$yearco <- munich_rent_index_01$yearc - mean(munich_rent_index_01$yearc)
poly_year_01 <- poly(munich_rent_index_01$yearco, 3)
munich_rent_index_01$yearco <- poly_year_01[, 1]
munich_rent_index_01$yearco2 <- poly_year_01[, 2]
munich_rent_index_01$yearco3 <- poly_year_01[, 3]
interaction_model <- lm(
rentsqm_euro ~ areainvc + yearco + yearco2 + yearco3 + year01 + nkitchen + pkitchen + nkitchen * year01 + pkitchen * year01,
data = munich_rent_index_01
)
summary(interaction_model)
Call:
lm(formula = rentsqm_euro ~ areainvc + yearco + yearco2 + yearco3 +
year01 + nkitchen + pkitchen + nkitchen * year01 + pkitchen *
year01, data = munich_rent_index_01)
Residuals:
Min 1Q Median 3Q Max
-7.6303 -1.2647 -0.1072 1.2414 10.5233
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.95424 0.03841 181.072 < 2e-16 ***
areainvc 123.71308 4.42630 27.950 < 2e-16 ***
yearco 49.42282 2.02589 24.396 < 2e-16 ***
yearco2 29.45781 2.02386 14.555 < 2e-16 ***
yearco3 -1.03886 1.97176 -0.527 0.598311
year01 -0.25174 0.06678 -3.770 0.000166 ***
nkitchen 0.91567 0.12172 7.523 6.43e-14 ***
pkitchen 1.09081 0.17849 6.111 1.07e-09 ***
year01:nkitchen 0.39826 0.20922 1.904 0.057033 .
year01:pkitchen 0.73511 0.32864 2.237 0.025346 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.963 on 4561 degrees of freedom
Multiple R-squared: 0.3397, Adjusted R-squared: 0.3384
F-statistic: 260.7 on 9 and 4561 DF, p-value: < 2.2e-16
Example 3.8 Munich Rent Index — Interaction Between Living Area and
Location
The model to adjust ius defined as:
\[
\mathbb{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot
\text{tlocation} + \beta_{2} \cdot \overline{\frac{1}{\text{area}}} +
\beta{3} \cdot \text{area} \cdot \text{tlocation}
\]
We use the dataset for average or top location only and change the
dummy variable \(\text{tlocation}\) to
\(0\) (average) and \(1\) (top). Finally, we visualize the effect
as described in the book.
par(mfrow = c(1, 2))
at_rent <- subset(munich_rent_index, location == 1 | location == 3)
at_rent$tlocation[at_rent$location == 1] <- 0
summary(at_rent)
rent rentsqm area yearc location
Min. : 54.92 Min. : 1.417 Min. : 20.00 Min. :1918 1:1794
1st Qu.: 314.62 1st Qu.: 5.276 1st Qu.: 50.00 1st Qu.:1954 2: 0
Median : 418.00 Median : 6.849 Median : 65.00 Median :1962 3: 78
Mean : 444.22 Mean : 7.007 Mean : 66.02 Mean :1959
3rd Qu.: 537.42 3rd Qu.: 8.652 3rd Qu.: 80.00 3rd Qu.:1972
Max. :1459.23 Max. :15.428 Max. :155.00 Max. :1997
bath kitchen cheating district yearco
Mode :logical Mode :logical Mode :logical 623 : 53 Min. :18.00
FALSE:1761 FALSE:1785 FALSE:148 1013 : 37 1st Qu.:54.00
TRUE :111 TRUE :87 TRUE :1724 713 : 32 Median :62.00
1132 : 31 Mean :59.36
1712 : 30 3rd Qu.:72.00
2024 : 30 Max. :97.00
(Other):1659
areainv yearcc yearcc2
Min. :-0.0105206 Min. :-0.030935 Min. :-0.0183373
1st Qu.:-0.0044722 1st Qu.:-0.001862 1st Qu.:-0.0147283
Median :-0.0015876 Median : 0.004598 Median :-0.0067317
Mean : 0.0002152 Mean : 0.002465 Mean :-0.0008428
3rd Qu.: 0.0030278 3rd Qu.: 0.012674 3rd Qu.: 0.0116794
Max. : 0.0330278 Max. : 0.032863 Max. : 0.0537875
yearcc3 glocation tlocation
Min. :-0.0186291 Min. :-1.0000 Min. :0.00000
1st Qu.:-0.0172309 1st Qu.:-1.0000 1st Qu.:0.00000
Median :-0.0056305 Median :-1.0000 Median :0.00000
Mean :-0.0004071 Mean :-0.9583 Mean :0.04167
3rd Qu.: 0.0096124 3rd Qu.:-1.0000 3rd Qu.:0.00000
Max. : 0.0588009 Max. : 0.0000 Max. :1.00000
area_location_model <- lm(
rentsqm ~ tlocation + areainv + area:tlocation,
data = at_rent
)
summary(area_location_model)
Call:
lm(formula = rentsqm ~ tlocation + areainv + area:tlocation,
data = at_rent)
Residuals:
Min 1Q Median 3Q Max
-7.2380 -1.4435 -0.1189 1.4788 7.1162
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.911e+00 4.967e-02 139.131 <2e-16 ***
tlocation 7.722e-01 7.280e-01 1.061 0.289
areainv 1.431e+02 7.434e+00 19.252 <2e-16 ***
tlocation:area 1.001e-02 8.612e-03 1.163 0.245
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.101 on 1868 degrees of freedom
Multiple R-squared: 0.1775, Adjusted R-squared: 0.1762
F-statistic: 134.4 on 3 and 1868 DF, p-value: < 2.2e-16
beta_al <- area_location_model$coefficients
f1_area <- function(area){
areainvc <- (1 / area) - mean(1 / area)
beta_al[3] * areainvc
}
f2_area <- function(area){
beta_al[4] * area
}
small_effect <- f1_area(area_seq)
big_effect <- beta_al[2] + f1_area(area_seq) + f2_area(area_seq)
ylim <- c(min(c(small_effect, big_effect)), max(c(small_effect, big_effect)))
plot(
area_seq,
small_effect,
main = 'area effects based on a linear interaction',
xlab = 'area in sqm',
ylab = 'area effects',
ylim = ylim,
type = 'l',
col = 'darkblue',
lwd = 2
)
lines(area_seq, big_effect, col = 'red', lwd = 2)
plot(
area_seq,
beta_al[2] + f2_area(area_seq),
main = 'varying effect of top location',
xlab = 'area in sqm',
ylab = 'Varying effect of top location',
type = 'l',
col = 'darkblue',
lwd = 2
)

Example 3.9 Orthogonal Design
Just for exemplify, we define the orthogonal process describe as an R
function:
\[
\tilde{x}^{j} = x^{j} - \widetilde{X}_{j} \left (
{\widetilde{X}_{j}}^{T} \widetilde{X}_{j} \right )^{-1}
{\widetilde{X}_{j}}^{T} x^{j}
\]
orthogonal_design <- function(x){
x_out <- cbind(x[, 1] - mean(x[, 1]), matrix(0, nrow = nrow(x), ncol = ncol(x) - 1))
for (i in 2:ncol(x)){
x_hat <- as.matrix(x_out[, 1:(i-1)])
x_inv <- solve(t(x_hat) %*% x_hat)
x_out[, i] <- x[, i] - x_hat %*% x_inv %*% t(x_hat) %*% x[, i]
}
x_out
}
We test this function on the polynomial of the \(\text{yearc}\) variable:
poly_test <- poly(munich_rent_index$yearc, 3, raw = TRUE)
orthogonal_test <- orthogonal_design(poly_test)
# And check it
c(
orthogonal_test[, 1] %*% orthogonal_test[, 2],
orthogonal_test[, 2] %*% orthogonal_test[, 3],
orthogonal_test[, 1] %*% orthogonal_test[, 3]
)
[1] 7.550556e-05 8.482947e+05 2.602290e+00
Why did it not result?
Example 3.10 Munich Rent Index — Comparison of Models Using \({R}^{2}\)
We compare the models: \(M1\), \(M2\), \(M3\), and \(M4\) using its coefficient of determination
\(R^{2}\)
coeff_deter <- data.frame(
`R Squared` = c(summary(m1)$r.squared, summary(m2)$r.squared, summary(m3)$r.squared, summary(m4)$r.squared),
row.names = c("M1", "M2", "M3", "M4")
)
coeff_deter
Example 3.11 Simple Linear Regression
For the model:
\[
y_{i} = \beta x_{i} + \epsilon_{i}
\]
We can verify:
\[
\left ( {X_{n}}^{T} X_{n} \right )^{-1} \to 0
\]
And:
\[
\max_{i=1,...,n} {x_{i}}^{T} \left ( {X_{n}}^{T} X_{n} \right )^{-1}
x_{i} \to 0 \text{ for } n \to \infty
\]
\(x_{i} = i\):
\[
\left ( {X_{n}}^{T} X_{n} \right )^{-1} = \left ( (1, 2, 3, \cdots, n)
\left ( \begin{matrix}
1 \\
2 \\
\vdots \\
n
\end{matrix} \right ) \right )^{-1} = \left ( 1 + 4 + \cdots + n^2
\right )^{-1} \to 0
\]
And:
\[
\max_{i=1,...,n} ( 1, 2, \cdots, n ) \left ( 1 + 4 + \cdots + n^2 \right
)^{-1} \left ( \begin{matrix}
1 \\
2 \\
\vdots \\
n
\end{matrix} \right ) \to 0 \text{ for } n \to \infty
\]
\(x_{i} = \frac{1}{i}\):
\[
\left ( {X_{n}}^{T} X_{n} \right )^{-1} = \left ( \left (1, 0.5, 0.33,
\cdots, \frac{1}{n} \right ) \left ( \begin{matrix}
1 \\
0.5 \\
\vdots \\
\frac{1}{n}
\end{matrix} \right ) \right )^{-1} = \left ( \sum_{i=1}^{n}
\frac{1}{i^{2}} \right )^{-1} \not{\to} 0
\]
And:
\[
\max_{i=1,...,n} \left ( 1, 0.5, \cdots, \frac{1}{n} \right ) \left (
\sum_{i = 1}^{n} \frac{1}{i^{2}} \right )^{-1} \left ( \begin{matrix}
1 \\
0.5 \\
\vdots \\
\frac{1}{n}
\end{matrix} \right ) \not{\to} 0 \text{ for } n \to \infty
\]
\(x_{i} =
\frac{1}{\sqrt{i}}\):
\[
\left ( {X_{n}}^{T} X_{n} \right )^{-1} = \left ( \sum_{i = 1}^{n}
\frac{1}{i} \right )^{-1} \to 0
\]
And:
\[
\max_{i=1,...,n} \left ( 1, \frac{1}{\sqrt{2}}, \cdots,
\frac{1}{\sqrt{n}} \right ) \left ( \sum_{i = 1}^{n} \frac{1}{i} \right
)^{-1} \begin{matrix}
1 \\
\frac{1}{\sqrt{2}} \\
\vdots \\
\frac{1}{\sqrt{n}}
\end{matrix} \to 0
\]
Example 3.12 Munich Rent Index — Hypothesis Testing
The regression model to adjust is:
\[
\begin{matrix}
\text{rentsqm}_{i} = & \beta_{0} + \beta_{1} \text{areainvc}_{i} +
\beta_{2} \text{yearco}_{i} + \beta_{3} \text{yearco2}_{i} + \beta_{4}
\text{yearco3}_{i} \\
& + \beta_{5} \text{nkitchen}_{i} + \beta_{6} \text{pkitchen}_{i} +
\beta_{7} \text{year01}_{i} + \epsilon_{i}
\end{matrix}
\]
hypothesis_model <- lm(
rentsqm_euro ~ areainvc + yearco + yearco2 + yearco3 + nkitchen + pkitchen + year01,
data = munich_rent_index_01
)
summary(hypothesis_model)
Call:
lm(formula = rentsqm_euro ~ areainvc + yearco + yearco2 + yearco3 +
nkitchen + pkitchen + year01, data = munich_rent_index_01)
Residuals:
Min 1Q Median 3Q Max
-7.674 -1.270 -0.113 1.242 10.479
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.93239 0.03757 184.539 < 2e-16 ***
areainvc 123.76987 4.42908 27.945 < 2e-16 ***
yearco 49.37274 2.02573 24.373 < 2e-16 ***
yearco2 29.49985 2.02515 14.567 < 2e-16 ***
yearco3 -0.88418 1.97229 -0.448 0.65396
nkitchen 1.04340 0.10151 10.279 < 2e-16 ***
pkitchen 1.30205 0.15226 8.552 < 2e-16 ***
year01 -0.18524 0.06213 -2.981 0.00288 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.964 on 4563 degrees of freedom
Multiple R-squared: 0.3385, Adjusted R-squared: 0.3375
F-statistic: 333.6 on 7 and 4563 DF, p-value: < 2.2e-16
So we want to test the hypothesis:
\[
\begin{matrix}
H_{0} : \beta_{7} = 0 & \text{against} & H_{1} : \beta_{7} \neq
0
\end{matrix}
\]
About the significance of the follow-up measure.
\[
\begin{matrix}
H_{0} : \left ( \begin{matrix}
\beta_{5} \\
\beta_{6}
\end{matrix} \right ) = \left ( \begin{matrix}
0 \\
0
\end{matrix} \right ) & \text{against} & H_{1} : \left (
\begin{matrix}
\beta_{5} \\
\beta_{6}
\end{matrix} \right ) \neq \left ( \begin{matrix}
0 \\
0
\end{matrix} \right )
\end{matrix}
\]
About the significance of the kitchen variable. And a final test
about the significance of differentiate between these two type of
kitchens:
\[
\begin{matrix}
H_{0} : \beta_{5} - \beta_{6} = 0 & \text{against} & H_{1} :
\beta_{5} - \beta_{6} \neq 0
\end{matrix}
\]
Example 3.13 Munich Rent Index — Standard Output and Hypothesis
Tests
For the first test (\(H_{0} : \beta_{7} =
0\)), we calculate:
\[
t_{7} =
\frac{\hat{\beta}_{7}}{{\widehat{\text{Var}(\hat{\beta}_{7})}}^{1/2}}
\sim t_{1 - \alpha / 2} (n - p)
\]
The covariance matrix is:
cov_matrix <- vcov(hypothesis_model)
cov_matrix
(Intercept) areainvc yearco yearco2 yearco3
(Intercept) 0.001411211 0.006823735 0.004991278 0.005173670 -0.002333528
areainvc 0.006823735 19.616785356 -1.294964483 1.420604414 0.122987111
yearco 0.004991278 -1.294964483 4.103591466 0.046451741 -0.006404590
yearco2 0.005173670 1.420604414 0.046451741 4.101229176 0.027782090
yearco3 -0.002333528 0.122987111 -0.006404590 0.027782090 3.889943976
nkitchen -0.001093652 -0.091944737 -0.025946481 -0.026748293 0.006345756
pkitchen -0.001128930 0.022989833 -0.042249319 -0.049085309 -0.016014007
year01 -0.001270640 0.004137400 -0.002253660 -0.001730023 0.007205398
nkitchen pkitchen year01
(Intercept) -1.093652e-03 -0.0011289298 -1.270640e-03
areainvc -9.194474e-02 0.0229898326 4.137400e-03
yearco -2.594648e-02 -0.0422493193 -2.253660e-03
yearco2 -2.674829e-02 -0.0490853093 -1.730023e-03
yearco3 6.345756e-03 -0.0160140070 7.205398e-03
nkitchen 1.030419e-02 0.0014042193 5.682974e-05
pkitchen 1.404219e-03 0.0231824417 1.902243e-04
year01 5.682974e-05 0.0001902243 3.860040e-03
We check the \(\text{Var}(\hat{\beta}_{7})\) is the last
value.
var_7 <- cov_matrix[8, 8]
t_7 <- hypothesis_model$coefficients[8] / sqrt(var_7)
t_7
year01
-2.981496
And this coincides with the value given in the
summary.
For the second test, we compute the \(F\) value as:
\[
F = \frac{1}{r} {\hat{\beta}}^{T} \widehat{\text{Cov} \left (
\hat{\beta} \right )}^{-1} \hat{\beta} \sim F_{1, n - p}
\]
With the \(\hat{\beta}\) define in
the test \(\left ( \begin{matrix}
\hat{\beta_{5}} \\
\hat{\beta_{6}}
\end{matrix} \right )\), then \(r=2\) and \(n - p
= n - 8\):
df <- nrow(munich_rent_index_01) - 8
r <- 2
beta_hat <- as.matrix(hypothesis_model$coefficients[6:7])
f_56 <- (1/2) * t(beta_hat) %*% solve(cov_matrix[6:7, 6:7]) %*% beta_hat
f_56
[,1]
[1,] 82.08307
We get expected \(F\):
f_crit <- qf(p = 0.95, r, df)
f_crit
[1] 2.9977
And we reject the null hypothesis.
The third and final test comparing \(\beta_{5}\) and \(\beta_{6}\) between each other. We
need:
\[
\widehat{\text{Var}(\hat{\beta}_{5} - \hat{\beta}_{6})} =
\widehat{\text{Var}(\hat{\beta}_{5})} +
\widehat{\text{Var}(\hat{\beta}_{6})} - 2 \cdot
\widehat{\text{Cov}(\hat{\beta}_{5}, \hat{\beta}_{6})}
\]
Using the \(\widehat{\text{Cov}(\hat{\beta})}\)
matrix:
cov_5_6 <- cov_matrix[6, 7]
var_5 <- cov_matrix[6, 6]
var_6 <- cov_matrix[7, 7]
var_5_6 <- var_5 + var_6 - 2 * cov_5_6
f_5_6 <- (hypothesis_model$coefficients[6] - hypothesis_model$coefficients[7])^2 / var_5_6
f_5_6
nkitchen
2.18071
And this \(F\) critical, using \(r=1\):
f_crit <- qf(p = 0.95, 1, df)
f_crit
[1] 3.843498
So, in this case, we do not reject the null hypothesis.
Example 3.14 Munich Rent Index — Confidence Intervals
The confidence interval for \(\beta_{7}\) is obtain using:
\[
\beta_{7} \pm t_{n - p} \left (1 - \alpha / 2 \right ) \cdot
\widehat{\text{Var}(\hat{\beta}_{7})}^{1/2}
\]
inter <- qt(1 - 0.05 / 2, df) * sqrt(var_7)
c(hypothesis_model$coefficients[8] - inter, hypothesis_model$coefficients[8] + inter)
year01 year01
-0.3070414 -0.0634347
Example 3.15 Munich Rent Index — Prediction Intervals
Using the prediction interval:
\[
{x_{0}}^{T} \cdot \hat{\beta} \pm t_{n-p}(1 - \alpha / 2) \hat{\sigma}
\left ( 1 + {x_{0}}^{T} \left (X^{T} X \right )^{1} x_{0} \right )^{1/2}
\]
We can obtain the \(\hat{\sigma}\)
directly from the model or using:
\[
(n - p) \hat{\sigma}^2 = \epsilon^{T} \epsilon
\]
sigma_model <- summary(hypothesis_model)$sigma
sigma_2 <- t(as.matrix(hypothesis_model$residuals)) %*% as.matrix(hypothesis_model$residuals) / (nrow(munich_rent_index_01) - 8)
sigma <- sqrt(sigma_2)
c(sigma_model, sigma)
[1] 1.964112 1.964112
We are also going to use the design matrix:
design_matrix <- as.matrix(hypothesis_model$model)
design_matrix[, 1] <- 1
colnames(design_matrix)[1] <- "1"
head(design_matrix)
1 areainvc yearco yearco2 yearco3 nkitchen pkitchen
1 1 0.0116672025 -0.011568466 -0.010460264 0.030118199 0 0
2 1 -0.0072887975 -0.011568466 -0.010460264 0.030118199 0 0
3 1 0.0175786025 0.009594692 -0.004320479 -0.013940551 0 0
4 1 0.0087368025 0.010256041 -0.003177207 -0.014569233 0 0
5 1 -0.0065948975 0.018853574 0.016932467 -0.005009151 0 0
6 1 -0.0007751975 0.003642554 -0.012015191 -0.002877678 0 0
year01
1 0
2 0
3 0
4 0
5 0
6 0
area_seq <- seq(min(munich_rent_index_01$area), max(munich_rent_index_01$area), length.out = 100)
areainvc <- (1 / area_seq) - mean(1 / area_seq)
betam <- hypothesis_model$coefficients
yearcc <- munich_rent_index_01[munich_rent_index_01$yearc == 1918, ][1, ]
# Three last terms were added for completeness
constant <- betam[1] + betam[3] * yearcc$yearco + betam[4] * yearcc$yearco2 + betam[5] * yearcc$yearco3 + betam[6] * 0 + betam[7] * 0 + betam[8] * 0
plot(
munich_rent_index_01$area,
munich_rent_index_01$rentsqm_euro,
xlab = 'area in sqm',
ylab = 'estimated rent per sqm'
)
beta_1 <- hypothesis_model$coefficients[2]
lines(area_seq, constant + beta_1 * areainvc, col = 'red', lwd = 2)
conf_inter <- qt(1 - 0.05 / 2, df) * sqrt(cov_matrix[2, 2])
beta_conf <- c(beta_1 - conf_inter, beta_1 + conf_inter)
lines(area_seq, constant + beta_conf[1] * areainvc, col = 'darkblue', lwd = 2)
lines(area_seq, constant + beta_conf[2] * areainvc, col = 'darkblue', lwd = 2)
x_o <- as.matrix(cbind(
1,
areainvc,
yearcc$yearco,
yearcc$yearco2,
yearcc$yearco3,
0,
0,
0
))
pred_inter <- NULL
for (i in seq_len(nrow(x_o)))
x_o_i <- as.matrix(x_o[i, ])
pred_inter <- c(pred_inter, qt(1 - 0.05 / 2, df) * sigma * sqrt(
1 + (
t(x_o_i) %*% solve(t(design_matrix)%*%design_matrix) %*% x_o_i
))
)
lines(area_seq, constant + beta_conf[1] * areainvc - pred_inter, col = 'gray', lwd = 2)
lines(area_seq, constant + beta_conf[2] * areainvc + pred_inter, col = 'gray', lwd = 2)

This can also be done using predict:
x_o <- as.data.frame(cbind(
areainvc,
yearcc$yearco,
yearcc$yearco2,
yearcc$yearco3,
0,
0,
0
))
colnames(x_o) <- colnames(hypothesis_model$model)[2:8]
confidence_interval <- predict(hypothesis_model, x_o, interval = "confidence")
plot(
munich_rent_index_01$area,
munich_rent_index_01$rentsqm_euro,
xlab = 'area in sqm',
ylab = 'estimated rent per sqm'
)
lines(area_seq, confidence_interval[, "fit"], col = 'red', lwd = 2)
lines(area_seq, confidence_interval[, "lwr"], col = 'darkblue', lwd = 2)
lines(area_seq, confidence_interval[, "upr"], col = 'darkblue', lwd = 2)
prediction_interval <- predict(hypothesis_model, x_o, interval = "prediction")
lines(area_seq, prediction_interval[, "lwr"], col = 'gray', lwd = 2)
lines(area_seq, prediction_interval[, "upr"], col = 'gray', lwd = 2)

Example 3.16 Polynomial Regression
Simulated data from the real model:
\[
y_{i} = -1 + 0.3 x_{i} + 0.4 {x_{i}}^{2} - 0.8 {x_{i}}^{3} +
\epsilon_{i} \text{ with } \epsilon_{i} \sim N(0, {0.07}^{2})
\]
par(mfrow = c(3, 2))
x_seq <- seq(0, 1, length.out = 50)
training_data <- data.frame(
x = x_seq,
y = -1 + 0.3 * x_seq + 0.4 * x_seq ^ 2 - 0.8 * x_seq ^ 3 + rnorm(50, 0, 0.07)
)
validation_data <- data.frame(
x = x_seq,
y = -1 + 0.3 * x_seq + 0.4 * x_seq ^ 2 - 0.8 * x_seq ^ 3 + rnorm(50, 0, 0.07)
)
plot_training_data <- function(title) {
plot(
training_data,
main = title,
xlab = 'x',
ylab = 'y'
)
}
plot_training_data('training_data')
plot(
validation_data,
main = 'validation_data',
xlab = 'x',
ylab = 'y'
)
adjust_polynomial <- function(l) {
model <- "y ~ x"
if (l != 1) {
for (i in 2:l){
model <- paste0(model, " + I(x^", i, ")")
}
}
lm(model, data = training_data)
}
polynomial_models <- lapply(1:9, adjust_polynomial)
plot_training_data('regression line')
lines(x_seq, predict(polynomial_models[1][[1]]), col = 'red', lwd = 2)
plot_training_data('polynomial regression with l=2')
lines(x_seq, predict(polynomial_models[2][[1]]), col = 'red', lwd = 2)
plot_training_data('polynomial regression with l=5')
lines(x_seq, predict(polynomial_models[5][[1]]), col = 'red', lwd = 2)
mean_square_error <- function(model, data){
(1 / 50) * sum((data$y - predict(model, new_data = data)) ^ 2)
}
mse_training_data <- unlist(lapply(polynomial_models, mean_square_error, data = training_data))
mse_validation_data <- unlist(lapply(polynomial_models, mean_square_error, data = validation_data))
plot(
seq(1, 9, 1),
mse_training_data,
ylim = c(min(mse_training_data, mse_validation_data), max(mse_training_data, mse_validation_data)),
main = 'MSE for training and validation data',
xlab = 'degree of polynomial',
ylab = 'MSE',
type = 'l',
col = 'darkblue',
lwd = 2
)
lines(seq(1, 9, 1), mse_validation_data, col = 'red', lwd = 2)

Example 3.17 Correlated Covariates
Simulate the data as following:
\[
x_{1} \sim U(0, 1) \wedge x_{3} \sim U(0, 1)
\]
\[
x_{2} = x_{1} + u \text{ with } u \sim U(0, 1)
\]
\[
y \sim N(-1 + 0.3 x_{1} + 0.2 x_{3}, {0.2}^{2})
\]
n <- 150
x1 <- runif(n, 0, 1)
x3 <- runif(n, 0, 1)
u <- runif(n, 0, 1)
x2 <- x1 + u
y <- rnorm(n, mean = -1 + 0.3 * x1 + 0.2 * x3, 0.2)
corr_cov_data <- data.frame(
y = y,
x1 = x1,
x2 = x2,
x3 = x3
)
pairs(corr_cov_data)

Adjust the model with \(x_{1}\),
\(x_{2}\) and \(x_{3}\):
wrong_model <- lm(y ~ x1 + x2 + x3, data = corr_cov_data)
summary(wrong_model)
Call:
lm(formula = y ~ x1 + x2 + x3, data = corr_cov_data)
Residuals:
Min 1Q Median 3Q Max
-0.45604 -0.12357 0.00926 0.11418 0.51568
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.983404 0.049308 -19.944 < 2e-16 ***
x1 0.308749 0.077842 3.966 0.000114 ***
x2 -0.003877 0.054825 -0.071 0.943728
x3 0.138695 0.051768 2.679 0.008228 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1858 on 146 degrees of freedom
Multiple R-squared: 0.2057, Adjusted R-squared: 0.1894
F-statistic: 12.6 on 3 and 146 DF, p-value: 2.252e-07
Adjust the model with \(x_{1}\) and
\(x_{3}\):
correct_model <- lm(y ~ x1 + x3, data = corr_cov_data)
summary(correct_model)
Call:
lm(formula = y ~ x1 + x3, data = corr_cov_data)
Residuals:
Min 1Q Median 3Q Max
-0.45722 -0.12438 0.00888 0.11348 0.51427
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.98526 0.04160 -23.685 < 2e-16 ***
x1 0.30469 0.05242 5.812 3.69e-08 ***
x3 0.13868 0.05159 2.688 0.00802 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.1852 on 147 degrees of freedom
Multiple R-squared: 0.2057, Adjusted R-squared: 0.1949
F-statistic: 19.03 on 2 and 147 DF, p-value: 4.465e-08
Example 3.18 Polynomial Regression — Model Choice with AIC
We calculate AIC using:
\[
AIC = n \cdot \log(\hat{\sigma}^{2}) + 2 \left ( \left | M \right | + 1
\right )
\]
aic <- function(model){
m <- ncol(model$model) - 1
n <- nrow(model$model)
n * log(summary(model)$sigma^2) + 2 * (m + 1)
}
aic_values <- unlist(lapply(polynomial_models, aic))
plot(
aic_values,
main = 'AIC as a function of degree of polynomials',
xlab = 'degrees of polynomial',
ylab = 'aic',
type = 'l',
col = 'darkblue',
lwd = 2
)

As always, this can also be done using R:
aic <- function(model){
AIC(model)
}
aic_values <- unlist(lapply(polynomial_models, aic))
plot(
aic_values,
main = 'AIC as a function of degree of polynomials',
xlab = 'degrees of polynomial',
ylab = 'aic',
type = 'l',
col = 'darkblue',
lwd = 2
)

Example 3.19 Prices of Used Cars — Model Choice
As for all dataset, we import the “prices of used cars”
dataset from the official page of the
dataset:
cars_url <- "https://www.uni-goettingen.de/de/document/download/062cadfda1c4c295f9460b49c7f5799e.raw/golffull.raw"
used_cars <- read.table(
url(cars_url),
header = 1,
colClasses = c(
"numeric", "integer", "numeric",
"integer", "factor", "factor",
"numeric", "numeric", "numeric",
"numeric", "numeric", "numeric"
)
)
# Convert to logical
used_cars$extras1 <- used_cars$extras1 == 1
used_cars$extras2 <- used_cars$extras2 == 1
summary(used_cars)
price age kilometer TIA
Min. :1.450 Min. : 65.00 Min. : 10.0 Min. : 0.00
1st Qu.:2.450 1st Qu.: 99.75 1st Qu.:107.0 1st Qu.:11.00
Median :3.100 Median :115.00 Median :130.8 Median :19.00
Mean :3.397 Mean :113.19 Mean :134.7 Mean :16.98
3rd Qu.:3.960 3rd Qu.:129.00 3rd Qu.:167.5 3rd Qu.:24.00
Max. :7.300 Max. :142.00 Max. :250.0 Max. :26.00
extras1 extras2 kilometerop1 kilometerop2
Mode :logical Mode :logical Min. :-2.79773 Min. :-0.7411
FALSE:54 FALSE:43 1st Qu.:-0.62208 1st Qu.:-0.6540
TRUE :118 TRUE :129 Median :-0.08938 Median :-0.3941
Mean : 0.00000 Mean : 0.0000
3rd Qu.: 0.73491 3rd Qu.: 0.2853
Max. : 2.58534 Max. : 5.3033
kilometerop3 ageop1 ageop2 ageop3
Min. :-6.9890 Min. :-2.54234 Min. :-1.0399 Min. :-5.6883
1st Qu.:-0.6275 1st Qu.:-0.70912 1st Qu.:-0.8309 1st Qu.:-0.7110
Median : 0.1185 Median : 0.09539 Median :-0.2932 Median :-0.1657
Mean : 0.0000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
3rd Qu.: 0.6005 3rd Qu.: 0.83395 3rd Qu.: 0.5750 3rd Qu.: 0.8218
Max. : 4.4942 Max. : 1.51976 Max. : 4.2063 Max. : 2.3620
---
title: "Classical Linear Model"
output: html_notebook
---

```{r}
set.seed(123)
```

## Homoscedastic Error Variances

For the visualization of homocedastic and hetercedastic variance, we simulate:

$$
y \sim N \left ( -1 + 2x, 1 \right )
$$

The funnel-shaped heterocedasticy is simulated using:

$$
y \sim N \left ( -1 + 2x, \left ( 0.1 + 0.3 \left ( x + 3 \right ) \right )^{2} \right )
$$

```{r, fig.width=10, fig.height=8, fig.align='center'}
par(mfrow = c(2, 2))

# Homecedastic variance
X <- seq(-2, 2, length.out = 300)

homecedastic_random <- function (x){
  rnorm(1, mean = -1 + 2 * x,  sd = 1)[1]
}

homo_y <- sapply(X, homecedastic_random)
homo_e <- (-1 + 2 * X) - homo_y

plot(X, homo_y, xlab='', ylab='', main='homocedastic variance')
abline(-1, 2, col = "red", lwd=2)
plot(X, homo_e, xlab='', ylab='', main='homocedastic variance errors')
abline(0, 0, col = "red", lwd=2)

heterocedastic_random <- function (x){
  rnorm(1, mean = -1 + 2 * x, sd = 0.1 + 0.3 * (x + 3))[1]
}

hetero_y <- sapply(X, heterocedastic_random)
hetero_e <- (-1 + 2 * X) - hetero_y

plot(X, hetero_y, xlab='', ylab='', main='funnel-shaped heterocedastic variance')
abline(-1, 2, col = "red", lwd=2)
plot(X, hetero_e, xlab='', ylab='', main='funnel-shaped heterocedastic variance errors')
abline(0, 0, col = "red", lwd=2)
```

## Example 3.1 Munich Rent Index—Heteroscedastic Variances

```{r, fig.width=10, fig.height=4, fig.align='center'}
par(mfrow = c(1, 2))

munich_rent_url <- "https://www.uni-goettingen.de/de/document/download/64c29c1b1fccb142cfa8f29a942a9e05.raw/rent99.raw"

munich_rent_index <- read.table(
  url(munich_rent_url),
  header = 1,
  colClasses = c(
    "numeric", "numeric", "numeric",
    "numeric", "factor", "factor",
    "factor", "factor", "factor"
  )
)

simple_reg <- lm(rent ~ area, data = munich_rent_index)

plot(munich_rent_index$area, munich_rent_index$rent, xlab = 'area in sqm', ylab = 'net rent in Euro')
abline(simple_reg, col = "red", lwd = 2)

predicted <- predict(simple_reg)

plot(munich_rent_index$area, predicted - munich_rent_index$rent, xlab = 'area in sqm', ylab = 'residuals')
abline(0, 0, col = "red", lwd = 2)
```

## Uncorrelated Errors

Simulated autocorrelated errors with positive correlation are generated using:

$$
y_{i} = -1 + 2 x_{i} + \epsilon_{i}
$$
$$
\text{Where } \epsilon_{i} = 0.9 \epsilon_{i - 1} + \mu_{i} \wedge \mu_{i} \sim N(0, 0.5^{2})
$$

```{r, fig.width=10, fig.height=4, fig.align='center'}
par(mfrow = c(1, 2))

n <- 100
X <- seq(-3, 3, length.out = n)
mu <- rnorm(n, mean = 0, sd = 0.5)

epsilon <- mu[1]
for(i in seq(2, n)){
  epsilon <- c(epsilon, 0.9 * epsilon[i - 1] + mu[i])
}

y_pred <- -1 + 2 * X + epsilon

plot(X, y_pred, xlab = 'x', ylab = '', main = 'observations and regression line')
abline(-1, 2, col = "red", lwd = 2)

plot(epsilon, xlab = 'i', ylab = '', main = 'time series of the errors')
abline(0, 0, col = "red", lwd = 2)
```

Simulated autocorrelated errors with negative correlation are generated using:

$$
y_{i} = -1 + 2 x_{i} + \epsilon_{i}
$$
$$
\text{Where } \epsilon_{i} = - 0.9 \epsilon_{i - 1} + \mu_{i} \wedge \mu_{i} \sim N(0, 0.5^{2})
$$

```{r, fig.width=10, fig.height=4, fig.align='center'}
par(mfrow = c(1, 2))

n <- 100
X <- seq(-3, 3, length.out = n)
mu <- rnorm(n, mean = 0, sd = 0.5)

epsilon <- mu[1]
for(i in seq(2, n)){
  epsilon <- c(epsilon, -0.9 * epsilon[i - 1] + mu[i])
}

y_pred <- -1 + 2 * X + epsilon

plot(X, y_pred, xlab = 'x', ylab = '', main = 'observations and regression line')
abline(-1, 2, col = "red", lwd = 2)

plot(epsilon, xlab = 'i', ylab = '', main = 'time series of the errors')
abline(0, 0, col = "red", lwd = 2)
```

Mispecified model is simulated using:

$$
y_{i} = \sin(x_{i}) + x_{i} + \epsilon_{i}
$$
$$
\text{Where } \epsilon_{i} \sim N(0, 0.3^{2})
$$

```{r, fig.width=10, fig.height=8, fig.align='center'}
par(mfrow = c(2, 2))

n <- 100
X <- seq(-3, 3, length.out = n)
epsilon <- rnorm(n, mean = 0, sd = 0.3)

y <- sin(X) + X + epsilon

plot(X, y, xlab = '', ylab = '', main = 'observations and true function')
lines(X, sin(X) + X, col = "red", lwd = 2)

linear_model <- lm(y ~ X)

plot(X, y, xlab = '', ylab = '', main = 'observations and regression line')
abline(linear_model, col = "red", lwd = 2)

y_pred <- predict(linear_model, data = X)
res <- y_pred - y
plot(X, res, xlab = '', ylab = '', main = 'residuals')
abline(0, 0, col = "red", lwd = 2)
```

## Additivity of Errors

We simulate data using:

$$
y_{i} = \exp(1 + x_{i1} - x_{i2} + \epsilon_{i})
$$

With:
$$
\epsilon_{i} \sim N \left ( 0, {0.4}^{2} \right )
$$

We plot this model:

```{r, fig.width=10, fig.height=8, fig.align='center'}
par(mfrow = c(2, 2))

n <- 50

x1 <- seq(0, 3, length.out = n)
x2 <- seq(0, 3, length.out = n)
x_grid <- expand.grid(x1 = x1, x2 = x2)
epsilon <- rnorm(n^2, mean = 0, sd = 0.4)
y <- exp(1 + x_grid$x1 - x_grid$x2 + epsilon)

plot(
  x_grid$x1, y,
  main = 'scatter plot: y versus x1',
  xlab = 'x1',
  ylab = 'y'
)
plot(
  x_grid$x2, y,
  main = 'scatter plot: y versus x2',
  xlab = 'x2',
  ylab = 'y'
)

plot(
  x_grid$x1, log(y),
  main = 'scatter plot: log(y) versus x1',
  xlab = 'x1',
  ylab = 'log(y)'
)
plot(
  x_grid$x2, log(y),
  main = 'scatter plot: log(y) versus x2',
  xlab = 'x2',
  ylab = 'log(y)'
)
```

## Example 3.2 Supermarket Scanner Data

Data not available online.

## Example 3.3 Munich Rent Index — Variable Transformation

Importing data using script:

```{r}
source("import_data/munich_rent_index.R")
```

We do the regression model:

$$
\text{rentsqm}_{i} = \beta_{0} + \beta_{1} \cdot f(\text{area}_{i}) + \epsilon_{i}
$$

With both $f(\cdot)$:

$$
f(\text{area}_{i}) = \text{area}_{i}
$$

For linear model and:

$$
f(\text{area}_{i}) = \frac{1}{\text{area}_{i}}
$$

For future use we define those models:

**Model 1 ($M1$)**:

$$
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \text{area}
$$

**Model 2 ($M2$)**:

$$
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \frac{1}{\text{area}}
$$

For nonlinear relationship between $\text{rentsqm}$ and $\text{area}$. The plotted graph below are those two models with the left panels the linear regression without the transformation and the right panels with the inverse transformation.

```{r, fig.width=10, fig.height=12, fig.align='center'}
par(mfrow = c(3, 2))

m1 <- lm(rentsqm ~ area, data = munich_rent_index)
summary(m1)

m2 <- lm(rentsqm ~ I(1 / area), data = munich_rent_index)
summary(m2)

plot_rentsqm_area <- function(title) {
  plot(
    munich_rent_index$area,
    munich_rent_index$rentsqm,
    main = title,
    xlab = 'area in sqm',
    ylab = 'rent per sqm'
  )
}

seq_area <- seq(min(munich_rent_index$area), max(munich_rent_index$area), length.out = 100)

plot_rentsqm_area('rent per sqm vs. area')
m1_beta <- m1$coefficients
lines(seq_area, m1_beta[1] + m1_beta[2] * seq_area, col = 'red', lwd = 2)

plot_rentsqm_area('f(area) = 1/area')
m2_beta <- m2$coefficients
lines(seq_area, m2_beta[1] + m2_beta[2] * 1 / seq_area, col = 'red', lwd = 2)

plot_residuals <- function(model) {
  plot(
    munich_rent_index$area,
    model$residuals,
    main = 'residuals',
    xlab = 'area in sqm',
    ylab = 'residuals'
  )
  abline(0, 0, col = 'red', lwd = 2)
}

plot_residuals(m1)
plot_residuals(m2)

plot_av_residuals <- function(model) {
  av_residuals <- tapply(
    model$residuals, munich_rent_index$area,
    mean
  )
  plot(
    unique(munich_rent_index$area),
    av_residuals,
    main = 'average residuals',
    xlab = 'area in sqm',
    ylab = 'residuals'
  )
  abline(0, 0, col = 'red', lwd = 2)
}

plot_av_residuals(m1)
plot_av_residuals(m2)
```


## Example 3.4 Munich Rent Index — Polynomial Regression

For polynomial regressions we adjust two different model. Of second-order (**Model 3 ($M3$)**):

$$
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \text{area} + \beta_{2} \text{area}^{2}
$$

And third-order (**Model 4 ($M4$)**):

$$
\mathbb{E}(\text{rentqsm}) = \beta_{0} + \beta_{1} \text{area} + \beta_{2} \text{area}^{2} + \beta_{3} \text{area}^{3}
$$

```{r, fig.width=10, fig.height=12, fig.align='center'}
par(mfrow = c(2, 2))

m3 <- lm(rentsqm ~ area + I(area^2), data = munich_rent_index)
summary(m3)

m4 <- lm(rentsqm ~ area + I(area^2) + I(area^3), data = munich_rent_index)
summary(m4)

plot_rentsqm_area('rent per sqm vs. area')
m3_beta <- m3$coefficients
lines(seq_area, m3_beta[1] + m3_beta[2] * seq_area + m3_beta[3] * seq_area ^ 2, col = 'red', lwd = 2)

plot_rentsqm_area('f(area) = 1/area')
m4_beta <- m4$coefficients
lines(seq_area, m4_beta[1] + m4_beta[2] * seq_area + m4_beta[3] * seq_area ^ 2 + m4_beta[4] * seq_area ^ 3, col = 'red', lwd = 2)

plot_residuals(m3)
plot_residuals(m4)
```

## Example 3.5 Munich Rent Index — Additive Models

We define the following aditive models. Additive model 1:

$$
\mathcal{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot \frac{1}{\text{area}_{i}} + \beta_{2} \cdot \text{yearc}_{i}
$$

And a second one with $\text{yearc}$ as a polinomial of degree 3:

$$
\mathcal{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot \frac{1}{\text{area}_{i}} + \beta_{2} \cdot \text{yearc}_{i} + \beta_{3} \cdot {\text{yearc}_{i}}^{2} + \beta_{4} \cdot {\text{yearc}_{i}}^{3}
$$

```{r}
additive_m1 <- lm(rentsqm ~ I(1/area) + yearc, data = munich_rent_index)
summary(additive_m1)

additive_m2 <- lm(rentsqm ~ I(1/area) + yearc + I(yearc ^ 2)+ I(yearc ^ 3), data = munich_rent_index)
summary(additive_m2)
```

A third model is specified using the truncated year of construction ($\text{yearc} - 1900$):

```{r}
munich_rent_index$yearco <- munich_rent_index$yearc - 1900
additive_m3 <- lm(rentsqm ~ I(1/area) + yearco + I(yearco ^ 2)+ I(yearco ^ 3), data = munich_rent_index)
summary(additive_m3)
```

As in the book, we show the effect of year of construction in the polynomial of degree 3:

```{r, fig.width=10, fig.height=4, fig.align='center'}
par(mfrow = c(1, 2))
beta_add_m2 <- additive_m2$coefficients

yearc_seq <- seq(min(munich_rent_index$yearc), max(munich_rent_index$yearc), length.out = 100)

plot_effect <- function(beta, year_seq, title){
  plot(
    year_seq,
    beta[3] * year_seq + beta[4] * year_seq^2 + beta[5] * year_seq^3,
    main = title,
    xlab = 'year of construction',
    ylab = 'effect of year of construction',
    type = 'l',
    col = 'darkblue',
    lwd = 2
  )
}

plot_effect(beta_add_m2, yearc_seq, 'effect of year of construction')

beta_add_m3 <- additive_m3$coefficients
truncate_yearc_seq <- seq(min(munich_rent_index$yearco), max(munich_rent_index$yearco), length.out = 100)
plot_effect(beta_add_m3, truncate_yearc_seq, 'effect of year of construction, coded from 18 to 96')
```

A new model is defined using the orthogonal polynomial of year of construction:

```{r}
munich_rent_index$areainv <- (1 / munich_rent_index$area) - mean(1 / munich_rent_index$area)
poly_year <- poly(munich_rent_index$yearco - mean(munich_rent_index$yearco), 3)
munich_rent_index$yearcc <- poly_year[, 1]
munich_rent_index$yearcc2 <- poly_year[, 2]
munich_rent_index$yearcc3 <- poly_year[, 3]
additive_m4 <- lm(rentsqm ~ areainv + yearcc + yearcc2 + yearcc3, data = munich_rent_index)
summary(additive_m4)
```

Now we calculate the partial residuals for $\text{area}$ and $\text{yearc}$ define by:

$$
\hat{\epsilon}_{\text{area}, i} = \text{rentsqm}_{i} - \hat{\beta_{0}} - \hat{f}_{2}(\text{yearc}_{i})
$$

With the effect $f(\text{yearc}_{i})$:

$$
\hat{f}_{2}(\text{yearc}) = \beta_{2} \cdot \text{yearc}_{i} + \beta_{3} \cdot {\text{yearc}_{i}}^{2} + \beta_{4} \cdot {\text{yearc}_{i}}^{3}
$$

And:

$$
\hat{\epsilon}_{\text{yearc}, i} = \text{rentsqm}_{i} - \hat{\beta_{0}} - \hat{f}_{1}(\text{area}_{i})
$$

With the effect $f(\text{yearc}_{i})$:

$$
\hat{f}_{1}(\text{yearc}) = \beta_{1} \cdot \frac{1}{\text{area}_{i}}
$$

```{r, fig.width=10, fig.height=4, fig.align='center'}
par(mfrow = c(1, 2))

beta_add_m4 <- additive_m4$coefficients

area_seq <- seq(min(munich_rent_index$area), max(munich_rent_index$area), length.out = 100)

area_effect <- function(area){
  inv_area <- 1 / area
  cent_area <- inv_area - mean(inv_area)
  beta_add_m4[2] * cent_area
}

yearc_effect <- function(yearc){
  yearc_center <- yearc - mean(yearc)
  poly_yearc <- poly(yearc_center, 3)
  beta_add_m4[3] * poly_yearc[, 1] + beta_add_m4[4] * poly_yearc[, 2] + beta_add_m4[5] * poly_yearc[, 3]
}

residual_area <- munich_rent_index$rentsqm - beta_add_m4[1] - yearc_effect(munich_rent_index$yearc)
plot(
  munich_rent_index$area,
  residual_area,
  main = 'effect of area',
  xlab = 'area in sqm',
  ylab = 'effect of area / partial residuals'
)
lines(area_seq, area_effect(area_seq), col = 'red', lwd = 2)

residual_year <- munich_rent_index$rentsqm - beta_add_m4[1] - area_effect(munich_rent_index$area)
plot(
  munich_rent_index$yearc,
  residual_year,
  main = 'effect of year of construction',
  xlab = 'year of construction',
  ylab = 'effect of year of construction / partial residuals'
)
lines(yearc_seq, yearc_effect(yearc_seq), col = 'red', lwd = 2)
```

## Example 3.6 Munich Rent Index — Effect Coding

We adjust the regression model with the coding values of location:

$$
\mathbb{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot \text{glocation}_{1} + \beta_{2} \cdot \text{tlocation}_{1}
$$

```{r}
munich_rent_index$glocation <- 0
munich_rent_index$glocation[munich_rent_index$location == 2] <- 1
munich_rent_index$glocation[munich_rent_index$location == 1] <- -1

munich_rent_index$tlocation <- 0
munich_rent_index$tlocation[munich_rent_index$location == 3] <- 1
munich_rent_index$tlocation[munich_rent_index$location == 1] <- -1

cod_model <- lm(
  rentsqm ~ glocation + tlocation,
  data = munich_rent_index
)
summary(cod_model)
```

## Example 3.7 Munich Rent Index — Interaction with Quality of Kitchen

We import the new updated model of 2001, available at [official page of the dataset](https://www.uni-goettingen.de/en/551625.html) as all the datasets:

```{r}
munich_rent_url_01 <- "https://www.uni-goettingen.de/de/document/download/e00d039e9c1f28ab404b27b0b14931f1.raw/rent98_00.raw"

munich_rent_index_01 <- read.table(
  url(munich_rent_url_01),
  header = 1,
  colClasses = c(
    "numeric", "numeric", "numeric",
    "numeric", "integer", "integer",
    "integer", "integer", "integer",
    "integer", "numeric", "numeric",
    "numeric"
  )
)

summary(munich_rent_index_01)
```

And we adjust the model:

$$
\begin{matrix}
  \widehat{\text{rentsqm}}_{i} = & \beta_{0} + \beta_{1} \text{areainvc}_{i} + \beta_{2} \text{yearco}_{i} + \beta_{3} \text{yearco2}_{i} + \beta_{4} \text{yearco3}_{i} \\
   & + \beta_{5} \text{year01}_{i} + \beta_{6} \text{nkitchen}_{i} + \beta_{7} \text{pkitchen}_{i} \\
   & + \beta_{8} \text{nkitchen}_{i} \cdot \text{year01}_{i} + \beta_{9} \text{pkitchen}_{i} \cdot \text{year01}_{i}
\end{matrix}
$$

```{r}
munich_rent_index_01$areainvc <- munich_rent_index_01$invarea - mean(munich_rent_index_01$invarea)
munich_rent_index_01$yearco <- munich_rent_index_01$yearc - mean(munich_rent_index_01$yearc)
poly_year_01 <- poly(munich_rent_index_01$yearco, 3)
munich_rent_index_01$yearco <- poly_year_01[, 1]
munich_rent_index_01$yearco2 <- poly_year_01[, 2]
munich_rent_index_01$yearco3 <- poly_year_01[, 3]

interaction_model <- lm(
  rentsqm_euro ~ areainvc + yearco + yearco2 + yearco3 + year01 + nkitchen + pkitchen + nkitchen * year01 + pkitchen * year01,
  data = munich_rent_index_01
)

summary(interaction_model)
```

## Example 3.8 Munich Rent Index — Interaction Between Living Area and Location

The model to adjust ius defined as:

$$
\mathbb{E}(\text{rentsqm}_{i}) = \beta_{0} + \beta_{1} \cdot \text{tlocation} + \beta_{2} \cdot \overline{\frac{1}{\text{area}}} + \beta{3} \cdot \text{area} \cdot \text{tlocation}
$$

We use the dataset for average or top location only and change the dummy variable $\text{tlocation}$ to $0$ (average) and $1$ (top). Finally, we visualize the effect as described in the book.

```{r, fig.width=10, fig.height=4, fig.align='center'}
par(mfrow = c(1, 2))
at_rent <- subset(munich_rent_index, location == 1 | location == 3)
at_rent$tlocation[at_rent$location == 1] <- 0
summary(at_rent)

area_location_model <- lm(
  rentsqm ~ tlocation + areainv + area:tlocation,
  data = at_rent
)
summary(area_location_model)

beta_al <- area_location_model$coefficients

f1_area <- function(area){
  areainvc <- (1 / area) - mean(1 / area)
  beta_al[3] * areainvc
}

f2_area <- function(area){
  beta_al[4] * area
}

small_effect <- f1_area(area_seq)
big_effect <- beta_al[2] + f1_area(area_seq) + f2_area(area_seq)

ylim <- c(min(c(small_effect, big_effect)), max(c(small_effect, big_effect)))

plot(
  area_seq,
  small_effect,
  main = 'area effects based on a linear interaction',
  xlab = 'area in sqm',
  ylab = 'area effects',
  ylim = ylim,
  type = 'l',
  col = 'darkblue',
  lwd = 2
)
lines(area_seq, big_effect, col = 'red', lwd = 2)

plot(
  area_seq,
  beta_al[2] + f2_area(area_seq),
  main = 'varying effect of top location',
  xlab = 'area in sqm',
  ylab = 'Varying effect of top location',
  type = 'l',
  col = 'darkblue',
  lwd = 2
)
```

## Example 3.9 Orthogonal Design

Just for exemplify, we define the orthogonal process describe as an R function:

$$
\tilde{x}^{j} = x^{j} - \widetilde{X}_{j} \left ( {\widetilde{X}_{j}}^{T} \widetilde{X}_{j} \right )^{-1} {\widetilde{X}_{j}}^{T} x^{j}
$$

```{r}
orthogonal_design <- function(x){
  x_out <- cbind(x[, 1] - mean(x[, 1]), matrix(0, nrow = nrow(x), ncol = ncol(x) - 1))
  for (i in 2:ncol(x)){
    x_hat <- as.matrix(x_out[, 1:(i-1)])
    x_inv <- solve(t(x_hat) %*% x_hat)
    x_out[, i] <- x[, i] - x_hat %*% x_inv %*% t(x_hat) %*% x[, i]
  }
  x_out
}
```
We test this function on the polynomial of the $\text{yearc}$ variable:

```{r}
poly_test <- poly(munich_rent_index$yearc, 3, raw = TRUE)
orthogonal_test <- orthogonal_design(poly_test)

# And check it
c(
  orthogonal_test[, 1] %*% orthogonal_test[, 2],
  orthogonal_test[, 2] %*% orthogonal_test[, 3],
  orthogonal_test[, 1] %*% orthogonal_test[, 3]
)
```

Why did it not result?

## Example 3.10 Munich Rent Index — Comparison of Models Using ${R}^{2}$

We compare the models: $M1$, $M2$, $M3$, and $M4$ using its coefficient of determination $R^{2}$

```{r}
coeff_deter <- data.frame(
  `R Squared` = c(summary(m1)$r.squared, summary(m2)$r.squared, summary(m3)$r.squared, summary(m4)$r.squared),
  row.names = c("M1", "M2", "M3", "M4")
)
coeff_deter
```

## Example 3.11 Simple Linear Regression

For the model:

$$
y_{i} = \beta x_{i} + \epsilon_{i}
$$

We can verify:

$$
\left ( {X_{n}}^{T} X_{n} \right )^{-1} \to 0
$$

And:

$$
\max_{i=1,...,n} {x_{i}}^{T} \left ( {X_{n}}^{T} X_{n} \right )^{-1} x_{i} \to 0 \text{ for } n \to \infty
$$

### $x_{i} = i$:

$$
\left ( {X_{n}}^{T} X_{n} \right )^{-1} = \left ( (1, 2, 3, \cdots, n) \left ( \begin{matrix}
1 \\
2 \\
\vdots  \\
n
\end{matrix} \right ) \right )^{-1} = \left ( 1 + 4 + \cdots + n^2 \right )^{-1} \to 0
$$

And:

$$
\max_{i=1,...,n} ( 1, 2, \cdots, n ) \left ( 1 + 4 + \cdots + n^2 \right )^{-1} \left ( \begin{matrix}
1 \\
2 \\
\vdots  \\
n
\end{matrix} \right ) \to 0 \text{ for } n \to \infty
$$

### $x_{i} = \frac{1}{i}$:

$$
\left ( {X_{n}}^{T} X_{n} \right )^{-1} = \left ( \left (1, 0.5, 0.33, \cdots, \frac{1}{n} \right ) \left ( \begin{matrix}
1 \\
0.5 \\
\vdots  \\
\frac{1}{n}
\end{matrix} \right ) \right )^{-1} = \left ( \sum_{i=1}^{n} \frac{1}{i^{2}} \right )^{-1} \not{\to} 0
$$

And:

$$
\max_{i=1,...,n} \left ( 1, 0.5, \cdots, \frac{1}{n} \right ) \left ( \sum_{i = 1}^{n} \frac{1}{i^{2}} \right )^{-1} \left ( \begin{matrix}
1 \\
0.5 \\
\vdots  \\
\frac{1}{n}
\end{matrix} \right ) \not{\to} 0 \text{ for } n \to \infty
$$

### $x_{i} = \frac{1}{\sqrt{i}}$:

$$
\left ( {X_{n}}^{T} X_{n} \right )^{-1} = \left ( \sum_{i = 1}^{n} \frac{1}{i} \right )^{-1} \to 0
$$

And:

$$
\max_{i=1,...,n} \left ( 1, \frac{1}{\sqrt{2}}, \cdots, \frac{1}{\sqrt{n}} \right ) \left ( \sum_{i = 1}^{n} \frac{1}{i} \right )^{-1} \begin{matrix}
1 \\
\frac{1}{\sqrt{2}} \\
\vdots \\
\frac{1}{\sqrt{n}}
\end{matrix} \to 0
$$

## Example 3.12 Munich Rent Index — Hypothesis Testing

The regression model to adjust is:

$$
\begin{matrix}
\text{rentsqm}_{i} = & \beta_{0} + \beta_{1} \text{areainvc}_{i} + \beta_{2} \text{yearco}_{i} + \beta_{3} \text{yearco2}_{i} + \beta_{4} \text{yearco3}_{i} \\
& + \beta_{5} \text{nkitchen}_{i} + \beta_{6} \text{pkitchen}_{i} + \beta_{7} \text{year01}_{i} + \epsilon_{i}
\end{matrix}
$$

```{r}
hypothesis_model <- lm(
  rentsqm_euro ~ areainvc + yearco + yearco2 + yearco3 + nkitchen + pkitchen + year01,
  data = munich_rent_index_01
)
summary(hypothesis_model)
```

So we want to test the hypothesis:

$$
\begin{matrix}
H_{0} : \beta_{7} = 0 & \text{against} & H_{1} : \beta_{7} \neq 0
\end{matrix}
$$

About the significance of the follow-up measure.

$$
\begin{matrix}
H_{0} : \left ( \begin{matrix}
\beta_{5} \\
\beta_{6}
\end{matrix} \right ) = \left ( \begin{matrix}
0 \\
0
\end{matrix} \right ) & \text{against} & H_{1} : \left ( \begin{matrix}
\beta_{5} \\
\beta_{6}
\end{matrix} \right ) \neq \left ( \begin{matrix}
0 \\
0
\end{matrix} \right )
\end{matrix}
$$

About the significance of the kitchen variable. And a final test about the significance of differentiate between these two type of kitchens:

$$
\begin{matrix}
H_{0} : \beta_{5} - \beta_{6} = 0 & \text{against} & H_{1} : \beta_{5} - \beta_{6} \neq 0
\end{matrix}
$$

## Example 3.13 Munich Rent Index — Standard Output and Hypothesis Tests

For the first test ($H_{0} : \beta_{7} = 0$), we calculate:

$$
t_{7} = \frac{\hat{\beta}_{7}}{{\widehat{\text{Var}(\hat{\beta}_{7})}}^{1/2}} \sim t_{1 - \alpha / 2} (n - p)
$$

The covariance matrix is:

```{r}
cov_matrix <- vcov(hypothesis_model)
cov_matrix
```

We check the $\text{Var}(\hat{\beta}_{7})$ is the last value.

```{r}
var_7 <- cov_matrix[8, 8]
t_7 <- hypothesis_model$coefficients[8] / sqrt(var_7)
t_7
```

And this coincides with the value given in the `summary`.

For the second test, we compute the $F$ value as:

$$
F = \frac{1}{r} {\hat{\beta}}^{T} \widehat{\text{Cov} \left ( \hat{\beta} \right )}^{-1} \hat{\beta} \sim F_{1, n - p}
$$

With the $\hat{\beta}$ define in the test $\left ( \begin{matrix}
\hat{\beta_{5}} \\
\hat{\beta_{6}}
\end{matrix} \right )$, then $r=2$ and $n - p = n - 8$:

```{r}
df <- nrow(munich_rent_index_01) - 8
r <- 2
beta_hat <- as.matrix(hypothesis_model$coefficients[6:7])
f_56 <- (1/2) * t(beta_hat) %*% solve(cov_matrix[6:7, 6:7]) %*% beta_hat
f_56
```

We get expected $F$:

```{r}
f_crit <- qf(p = 0.95, r, df)
f_crit
```

And we reject the null hypothesis.

The third and final test comparing $\beta_{5}$ and $\beta_{6}$ between each other. We need:

$$
\widehat{\text{Var}(\hat{\beta}_{5} - \hat{\beta}_{6})} = \widehat{\text{Var}(\hat{\beta}_{5})} + \widehat{\text{Var}(\hat{\beta}_{6})} - 2 \cdot \widehat{\text{Cov}(\hat{\beta}_{5}, \hat{\beta}_{6})}
$$

Using the $\widehat{\text{Cov}(\hat{\beta})}$ matrix:

```{r}
cov_5_6 <- cov_matrix[6, 7]
var_5 <- cov_matrix[6, 6]
var_6 <- cov_matrix[7, 7]
var_5_6 <- var_5 + var_6 - 2 * cov_5_6
f_5_6 <- (hypothesis_model$coefficients[6] - hypothesis_model$coefficients[7])^2 / var_5_6
f_5_6
```

And this $F$ critical, using $r=1$:

```{r}
f_crit <- qf(p = 0.95, 1, df)
f_crit
```

So, in this case, we do not reject the null hypothesis.

## Example 3.14 Munich Rent Index — Confidence Intervals

The confidence interval for $\beta_{7}$ is obtain using:

$$
\beta_{7} \pm t_{n - p} \left (1 - \alpha / 2 \right ) \cdot \widehat{\text{Var}(\hat{\beta}_{7})}^{1/2}
$$

```{r}
inter <- qt(1 - 0.05 / 2, df) * sqrt(var_7)
c(hypothesis_model$coefficients[8] - inter, hypothesis_model$coefficients[8] + inter)
```

## Example 3.15 Munich Rent Index — Prediction Intervals

Using the prediction interval:

$$
{x_{0}}^{T} \cdot \hat{\beta} \pm t_{n-p}(1 - \alpha / 2) \hat{\sigma} \left ( 1 + {x_{0}}^{T} \left (X^{T} X \right )^{1} x_{0} \right )^{1/2}
$$

We can obtain the $\hat{\sigma}$ directly from the model or using:

$$
(n - p) \hat{\sigma}^2 = \epsilon^{T} \epsilon
$$

```{r}
sigma_model <- summary(hypothesis_model)$sigma
sigma_2 <- t(as.matrix(hypothesis_model$residuals)) %*% as.matrix(hypothesis_model$residuals) / (nrow(munich_rent_index_01) - 8)
sigma <- sqrt(sigma_2)
c(sigma_model, sigma)
```

We are also going to use the design matrix:

```{r}
design_matrix <- as.matrix(hypothesis_model$model)
design_matrix[, 1] <- 1
colnames(design_matrix)[1] <- "1"
head(design_matrix)
```

```{r, fig.width=10, fig.height=6, fig.align='center'}
area_seq <- seq(min(munich_rent_index_01$area), max(munich_rent_index_01$area), length.out = 100)
areainvc <- (1 / area_seq) - mean(1 / area_seq)

betam <- hypothesis_model$coefficients
yearcc <- munich_rent_index_01[munich_rent_index_01$yearc == 1918, ][1, ]
# Three last terms were added for completeness
constant <- betam[1] + betam[3] * yearcc$yearco + betam[4] * yearcc$yearco2 + betam[5] * yearcc$yearco3 + betam[6] * 0 + betam[7] * 0 + betam[8] * 0

plot(
  munich_rent_index_01$area,
  munich_rent_index_01$rentsqm_euro,
  xlab = 'area in sqm',
  ylab = 'estimated rent per sqm'
)
beta_1 <- hypothesis_model$coefficients[2]
lines(area_seq, constant + beta_1 * areainvc, col = 'red', lwd = 2)

conf_inter <- qt(1 - 0.05 / 2, df) * sqrt(cov_matrix[2, 2])
beta_conf <- c(beta_1 - conf_inter, beta_1 + conf_inter)
lines(area_seq, constant + beta_conf[1] * areainvc, col = 'darkblue', lwd = 2)
lines(area_seq, constant + beta_conf[2] * areainvc, col = 'darkblue', lwd = 2)

x_o <- as.matrix(cbind(
  1,
  areainvc,
  yearcc$yearco,
  yearcc$yearco2,
  yearcc$yearco3,
  0,
  0,
  0
))

pred_inter <- NULL

for (i in seq_len(nrow(x_o)))
  x_o_i <- as.matrix(x_o[i, ])
  pred_inter <- c(pred_inter, qt(1 - 0.05 / 2, df) * sigma * sqrt(
    1 + (
      t(x_o_i) %*% solve(t(design_matrix)%*%design_matrix) %*% x_o_i
   ))
)

lines(area_seq, constant + beta_conf[1] * areainvc - pred_inter, col = 'gray', lwd = 2)
lines(area_seq, constant + beta_conf[2] * areainvc + pred_inter, col = 'gray', lwd = 2)
```

This can also be done using `predict`:

```{r, fig.width=10, fig.height=6, fig.align='center'}
x_o <- as.data.frame(cbind(
  areainvc,
  yearcc$yearco,
  yearcc$yearco2,
  yearcc$yearco3,
  0,
  0,
  0
))
colnames(x_o) <- colnames(hypothesis_model$model)[2:8]
confidence_interval <- predict(hypothesis_model, x_o, interval = "confidence")

plot(
    munich_rent_index_01$area,
    munich_rent_index_01$rentsqm_euro,
    xlab = 'area in sqm',
    ylab = 'estimated rent per sqm'
)
lines(area_seq, confidence_interval[, "fit"], col = 'red', lwd = 2)
lines(area_seq, confidence_interval[, "lwr"], col = 'darkblue', lwd = 2)
lines(area_seq, confidence_interval[, "upr"], col = 'darkblue', lwd = 2)

prediction_interval <- predict(hypothesis_model, x_o, interval = "prediction")
lines(area_seq, prediction_interval[, "lwr"], col = 'gray', lwd = 2)
lines(area_seq, prediction_interval[, "upr"], col = 'gray', lwd = 2)
```

## Example 3.16 Polynomial Regression

Simulated data from the real model:

$$
y_{i} = -1 + 0.3 x_{i} + 0.4 {x_{i}}^{2} - 0.8 {x_{i}}^{3} + \epsilon_{i} \text{ with } \epsilon_{i} \sim N(0, {0.07}^{2})
$$

```{r, fig.width=10, fig.height=12, fig.align='center'}
par(mfrow = c(3, 2))

x_seq <- seq(0, 1, length.out = 50)
training_data <- data.frame(
  x = x_seq,
  y = -1 + 0.3 * x_seq + 0.4 * x_seq ^ 2 - 0.8 * x_seq ^ 3 + rnorm(50, 0, 0.07)
)
validation_data <- data.frame(
  x = x_seq,
  y = -1 + 0.3 * x_seq + 0.4 * x_seq ^ 2 - 0.8 * x_seq ^ 3 + rnorm(50, 0, 0.07)
)

plot_training_data <- function(title) {
  plot(
    training_data,
    main = title,
    xlab = 'x',
    ylab = 'y'
  )
}

plot_training_data('training_data')
plot(
  validation_data,
  main = 'validation_data',
  xlab = 'x',
  ylab = 'y'
)

adjust_polynomial <- function(l) {
  model <- "y ~ x"
  if (l != 1) {
    for (i in 2:l){
      model <- paste0(model, " + I(x^", i, ")")
    }
  }
  lm(model, data = training_data)
}

polynomial_models <- lapply(1:9, adjust_polynomial)
plot_training_data('regression line')
lines(x_seq, predict(polynomial_models[1][[1]]), col = 'red', lwd = 2)

plot_training_data('polynomial regression with l=2')
lines(x_seq, predict(polynomial_models[2][[1]]), col = 'red', lwd = 2)

plot_training_data('polynomial regression with l=5')
lines(x_seq, predict(polynomial_models[5][[1]]), col = 'red', lwd = 2)

mean_square_error <- function(model, data){
  (1 / 50) * sum((data$y - predict(model, new_data = data)) ^ 2)
}

mse_training_data <- unlist(lapply(polynomial_models, mean_square_error, data = training_data))
mse_validation_data <- unlist(lapply(polynomial_models, mean_square_error, data = validation_data))

plot(
  seq(1, 9, 1),
  mse_training_data,
  ylim = c(min(mse_training_data, mse_validation_data), max(mse_training_data, mse_validation_data)),
  main = 'MSE for training and validation data',
  xlab = 'degree of polynomial',
  ylab = 'MSE',
  type = 'l',
  col = 'darkblue',
  lwd = 2
)
lines(seq(1, 9, 1), mse_validation_data, col = 'red', lwd = 2)
```

## Example 3.17 Correlated Covariates

Simulate the data as following:

$$
x_{1} \sim U(0, 1) \wedge x_{3} \sim U(0, 1)
$$

$$
x_{2} = x_{1} + u \text{ with } u \sim U(0, 1)
$$

$$
y \sim N(-1 + 0.3 x_{1} + 0.2 x_{3}, {0.2}^{2})
$$

```{r, fig.width=10, fig.height=10, fig.align='center'}
n <- 150

x1 <- runif(n, 0, 1)
x3 <- runif(n, 0, 1)
u <- runif(n, 0, 1)

x2 <- x1 + u

y <- rnorm(n, mean = -1 + 0.3 * x1 + 0.2 * x3, 0.2)

corr_cov_data <- data.frame(
  y = y,
  x1 = x1,
  x2 = x2,
  x3 = x3
)
pairs(corr_cov_data)
```

Adjust the model with $x_{1}$, $x_{2}$ and $x_{3}$:

```{r}
wrong_model <- lm(y ~ x1 + x2 + x3, data = corr_cov_data)
summary(wrong_model)
```

Adjust the model with $x_{1}$ and $x_{3}$:

```{r}
correct_model <- lm(y ~ x1 + x3, data = corr_cov_data)
summary(correct_model)
```

## Example 3.18 Polynomial Regression — Model Choice with AIC

We calculate AIC using:

$$
AIC = n \cdot \log(\hat{\sigma}^{2}) +  2 \left ( \left | M \right | + 1 \right )
$$

```{r, fig.width=5, fig.height=4, fig.align='center'}
aic <- function(model){
  m <- ncol(model$model) - 1
  n <- nrow(model$model)
  n * log(summary(model)$sigma^2) + 2 * (m + 1)
}

aic_values <- unlist(lapply(polynomial_models, aic))
plot(
  aic_values,
  main = 'AIC as a function of degree of polynomials',
  xlab = 'degrees of polynomial',
  ylab = 'aic',
  type = 'l',
  col = 'darkblue',
  lwd = 2
)
```

As always, this can also be done using R:

```{r, fig.width=5, fig.height=4, fig.align='center'}
aic <- function(model){
  AIC(model)
}

aic_values <- unlist(lapply(polynomial_models, aic))
plot(
        aic_values,
        main = 'AIC as a function of degree of polynomials',
        xlab = 'degrees of polynomial',
        ylab = 'aic',
        type = 'l',
        col = 'darkblue',
        lwd = 2
)
```

## Example 3.19 Prices of Used Cars — Model Choice

As for all dataset, we import the _"prices of used cars"_ dataset from the [official page of the dataset](https://www.uni-goettingen.de/en/551625.html):

```{r}
cars_url <- "https://www.uni-goettingen.de/de/document/download/062cadfda1c4c295f9460b49c7f5799e.raw/golffull.raw"

used_cars <- read.table(
  url(cars_url),
  header = 1,
  colClasses = c(
    "numeric", "integer", "numeric",
    "integer", "factor", "factor",
    "numeric", "numeric", "numeric",
    "numeric", "numeric", "numeric"
  )
)

# Convert to logical
used_cars$extras1 <- used_cars$extras1 == 1
used_cars$extras2 <- used_cars$extras2 == 1
summary(used_cars)
```
